Long-term-care insurance was supposed to help pay for nursing homes, assisted living and personal aides for tens of millions of Americans when they became unable to take care of themselves.

Now, though, the industry is in financial turmoil, causing misery for many of the 7.3 million people who own a long-term-care policy, equal to about a fifth of the U.S. population at least 65 years old. Steep rate increases that many policyholders never saw coming are confronting them with an awful choice: Come up with the money to pay…